Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers.
Denote by $n$ the number of numbers in Katie's list. Let $x_1, x_2, \ldots, x_n$ be the numbers in her list. We are given that
\[x_1 + x_2 + \dots + x_n = x_1^2 + x_2^2 + \dots + x_n^2.\]By Cauchy-Schwarz,
\[(1^2 + 1^2 + \dots + 1^2)(x_1^2 + x_2^2 + \dots + x_n^2) \ge (x_1 + x_2 + \dots + x_n)^2.\]Hence,
\[n(x_1^2 + x_2^2 + \dots + x_n^2) \ge (x_1 + x_2 + \dots + x_n)^2 = (x_1^2 + x_2^2 + \dots + x_n^2) \cdot (1^2 + 1^2 + \dots + 1^2),\]so $n \ge 1^2 + 1^2 + \dots + 1^2 = n.$Since $n(x_1^2 + x_2^2 + \dots + x_n^2) \ge (x_1^2 + x_2^2 + \dots + x_n^2) \cdot (1^2 + 1^2 + \dots + 1^2),$ all the terms of the sequences $1,2,-(n - 2),-1,-1$ must be equal. Thus,
\[1 = -1,\]or $0 = 2.$ We conclude that no such list exist, so Katie doesn't have any numbers on her list, and the maximum possible arithmetic mean is $\boxed{0}.$